Electrical network



May 21, 1929. 1'1. 5. HOYT 1,713,603

ELECTRICAL NETWORK Filed April 9, 1923 5 Sheets-Sheet 2 ATTORNEY May 21,1929. R. s. HOYT ELECTRICAL Nmwoax 5 Sheets-Sheet '3 Filed April 9 192501 RC IN VEN TOR E J: 170%? A TTORNE Y Patented May 21, 1929 UNITEDSTATES PATENT OFFICE.

RAY S. HOYT, OF RUTHERFORD, NEW JERSEY, ASSIGNOR TO AMERICAN TELEPHONEAND TELEGRAPH COMPANY, A CORPORATION OF NEW YORK.

ELECTRICAL NETWORK.

Application filed April 9,

with the requirements of special cases. An-

other object is to provide a network to simulate a smooth transmissionline. These and other objects of my invention will become apparent onconsideration of a limited number of embodiments thereof which I havechosen as examples for disclosurein the following specification, withthe understanding that the specification relates more particularlythereto and that the invention will be defined in the appended claims.

My improved networks for simulating transmission lines will find usefulapplication in telephone s stems, and the embodiments of the inventionerein disclosed as examples are designed more especially with referenceto use in suchsystems.

The transmission of alternating currents over any transmission linebetween specified terminal impedances depends only on the propagationconstant and the characteristm impedance of the line (at the particularfrequency contemplated). In th1s sense, then, the properties oftransmission lmes maybe classed broadly aspropagation characteristicsand impedance characteristics. In telephony we are primarily concernedwith the depcndence of these characteristics on thcfrequency, over thetele honic frequency range.

Prior to the application of telephone repeaters to telephone lines thepropagation characteristics of such lines were more important than theirimpedance characteristics, because the received energy de ended muchmore on the former than on the atter.

The application of the two-way telephone repeater greatly altered therelative importance of these two characteristics, decreasing the needfor high transmittlng efiiclency of a line but greatly increasing thedependence of the results on the impedance of the line. As well known,this is because the amplification to which a two-way repeater can 1923.Serial No. 630,941.

be set without singin or even without senous 1n ury to the inteligibility of the transmlsslon, de ends strictly on the degreeofimpedance-ba ance between the lines or between the lmes and theirbalancing-networks. In the case of the 21-type repeater the two linesmust have such impedances as to closely balance each other throughoutthe telephonic frequency range. In the case of the 22-t-ype repeater,which for long lines requiring more than one repeater is superior to the2l-type,

impedance networks are required for closely balancing the im edances ofthe two lines throughout the te ephonic frequency range. Such balancingnetworks are necessary also 1n connection with the so-called four-wirerepeater circuit.

Smooth lines are fundamental in telephonic transmission; for anytelephone line 15 either a simple smooth line, or a compound smoothline, or a periodically loaded line whose sections are themselves shortsmooth lmes. In an case the characteristic impedances of t e constituentsmooth-lines enter importantly into the impedance of the system.Moreover the characteristic impedance of the series type of periodicallyloaded line, at frequencies low relatively to its critical frequency, isclosely the same as the characteristic impedance of the correspondingsmooth line.

Of course, the impedance of any line could be simulated, as closely asdesired, by means of an artificial model constructed of many shortsections each having lumped constants; but such structures would be veryexpensive and very cumbersome. Compared with them the networks describedin this specification are very simple non-periodic structures that arerelatively inexpensive and are quite compact; yet the more precise ofthem have proved-t0 be adequate for simulating with high precision theimpedance of most types of smooth lines, while even the least precise(which are the simplest) suffice for many applications. Thespecification includes first approximation design-formulae and outlinesa supplementary semi-graphical method for arriving at the bestproportioning of the networks. A typical illustrative example is workedout at the last of the specificationpreceding the claims.

- employed so as to facilitate Referring to the drawings, Figures 1, 2,2", 3, 4, 8, and 9 are diagrams in Cartesian coordinates representingcertain functional relation s that are involved in the followingexplanation of the principles of my invention and its embodiments hereindisclosed. Fig. 5 is a diagram showing one elementof my improvednetwork; Fig. 5 is a symbolic diagram showing another element which maytake various forms and to which I shall refer as the excess-simulator;Fig. 5 is a diagram of a complete network comprising the elements ofFigs. 5 and 5"; Fig. 5 is a diagram of a line which is simulated by thenetwork of Fig. 5; Figs. 6, 6 and 6 correspond respectively with Figs. 55 and 5 except that the excess-simulator is represented by a condenser;Fig. 7 is a diagram showing an excess-simulator adapted for more precisesimulation than the condenser of Fig. 6"; Fig. 7 shows anexcess-simulator equivalent to Fig. 7 Fig. 10 shows the form taken byeither of the excess-simulators of Figs. 7 and 7 when a certainparameter D, hereinafter defined, takes the value 0; Fig. 10 shows theform taken by either of the excesssimulators of Figs. 7 a and 7 when theparameter D takes the value 1; Fig. 11. shows a complete networkcomprising the excess-simulator of Fig. 7; Fig. 11 shows a completenetwork comprising the excess-simulator of Fig. 7 Figs. 11 and 11 showtwo networks each equivalent to those of Figs. 11 and 11"; Fig. 12 showsthe form taken by any one of the networks of Figs. 11 11 ,'11 and 11when the parameter D takes the value 0; Fig. 12 shows the limiting formwhich will be taken by either of the networks of Figs. 11 and 11 whenthe parameter D takes the value 1; Fig. 12 is an equivalent for Fig.12"; Fig. 18 shows a modification specially adapted for low frequency,with the excess simulator shunted by a resistance; Fig. 13 shows asimilar modification with the shunt differentially disposed; Fig. 14?-shows a modified excess-simulator when the unmodified excess-simulatorhas the form shown in Fig. 7; Fig. 14 shows a modified excesssimulatorwhen the unmodified excess-simu lator has the form shown in Fig. 7 Fig.14 shows a network equivalent to the excess-simulator of Fig. 14*; Fig.14: shows a network equivalent to any one of Figs. 14*, M and 14; I

Fig. 15 is a modification for low frequency derived from Fig. 11 byaddin a shunt resistance; Fig. 15 is in an equiva ent network for Fig.15"; Fig. 15 has two resistances in parallel equivalent to the singleresistance S of Fig. 15; Fig. 15 is an equivalent network for 15; Fig.16 shows a modification.

for a leaky line;"and Fig. 17 shows a modification in which severalsimilar networks are greater accuracy of adjustment The networks hereindisclosed as embodiments of my invention are designed more particularlywith reference to simulation of smooth lines.

R, G, I and C denoting, as usual, the fundamental line constants(constants so far as variations of current and voltage are concerned),namely the resistance, leakance (leakage conductance), inductance, andcapacity, per unit length; m denoting 27r-tlII18S the frequency f; and zthe operator However, this form is neither the simplest nor the mostsignificant. For it involves separately the four quantites R, G, L and Cand is thus a function of not less than four variables (five if a) isregarded separately from L and 'G), whereas its value evidently dependson only the relative value of these quantities and hence must beexpressible as a function of only three independent vari- .ablesnamelythe ratios of any three of them to the fourth.

In deciding just what form or forms of expression to adopt for K weshall here be guided by the following practical considerations:

(A) In telephony we are chiefly interested in the dependan'ce of K onthe frequency f; or, stated more generally, in the dependance of somequantity. that is approximately'proportional to K on some quantity thatis ap-' proximately proportional to f.

The class of smooth lines is comprised between the following two ratherwide extremes, having very different characteristics:

(B) At one extreme are the large gauge open-wire lines, particularlywhen used at high frequencies. For them R is small rela tively to L, andG relatively to C and hence K is approximately or at least roughly equalto (C) At the other extreme are the small gauge cables, particularlywhen used at low frequencies. For them R is large relatively to L,though G is small relatively to C; and

hence K is approximately or at least roughly equal to lR/iwO.

(D) The line constants R,"L, G do not voice frequency range; and hencethey, or combinations of them, serve suitably as parameters.

(E) The leakance G, which is nearly alchange much with frequency over atleast the.

ways the least important of the four line constants, usually variesgreatly with the frequency and hence by itself does not serve verysuitably as a parameter. However, in a wide range of applications Gisapproximately or at least roughly proportional to the frequency; andthen a Suitable parameter is G/f or preferably G /wG. This is' true ofcables except at extremely low frequencies. It is at least roughly trueof open-wire lines at very high frequencies, such as carrierfrequencies,

but usually not at voice frequencies. For most lines the leakance G isusually approximately or at least roughly a linear function of thefrequency, namely G=.G +vf, where Gr is the leakance at f=0, and v isapproximately independent of the frequency. For cables, Gr is smallcompared with vf except at very low values of); but for open-wire linesG is usually not negligible except at high values of In the light ofthese considerations a study ofequation (1) suggests the employment ofthe quantities F, E, k, g, a, I) defined by the following six equations.Not all of these substitutions will be employed simultaneously, but itis convenient to set them all down here together.

Usually F or E will be treated astheindependent variable; and k, g, a, bas parameters.

It should perhaps here be emphasized that the approximations mentionedin the foregoing set of five considerations, to (E), are employed merelyas guides in the selection of the variables and parameters defined bythe above equations (2) to (7), and in the choice of the forms adoptedbelow for the formula for the characteristic impedance. Except where thecontrary is definitely indicated, the formulae that will be adopted forthe characteristic impedance are rigorously exact; though the variablesF and E are never exactly proportional tovthe frequency, and the parameters is, g, a, b are never exactly independent of the frequency. Ifthe independent variables were exactly proportional to the frequency andthe parameters were exactly independent of the frequency, the graphs ofthe formulae would by a mere change of scales exactly represent theimpedance as an explicit function of the frequency.

With particular regard to considerations (B) and (C) it will be foundconvenient to divide the further treatment of smooth lines into two mainparts, pertaining to open-wire lines and to cables respectively; andthen, in each of those parts, to present the impedance formulae in thetwo forms respectively most suitable for the cases where the leakance isapproximately proportional to the frequency line parameters and on thefrequency range involved, rather than on the physical form of the line;for any line at sufficiently higlrfrequeneies has the open-wire type ofcharacteristics, and at sufficiently low frequencies the cable type ofcharacteristics. With regard to the relative importance of thefundamental line constants R, G, L, C when the frequency range is thatof the voice, it may be said that for the open-wire type of lines L andC are of about equal importance, R of secondary, and G of tertiaryimportance; on the other hand, for the cable type, R and C are of aboutequal importance, L of secondary, and G of tertiary importance. Inillustrationof the above remarks it may be noted that smoothly loadedcables (unless loaded very lightly) have the open-wire type of characteristics; as have also periodically loaded cables at low frequencies.

Before proceeding to the separate treat ments of open-wire lines andcables it seems desirable to indicate, briefly, the general nature ofthe effect produced on the impedance by leakance.

The general efi'eot 0f Zeal'ance.

In ordinary telephone cables the leakance is so small that, except atvery low frequencies, the impedance of such cables is closely the sameas in the limiting case of no leakance; whence that limiting case may betaken as being a good approximation to the actual case. In open-wirelines leakance may be much larger than in cables, yet normally it issmall enough so that its effects on the impedance are slight, except atvery low frequencies, so that usually the limiting value of zeroleakanceis still a good approximation when calculating thecharacteristic impedance. However, during wet weather and inparticularly humid climates and locations the leakance in open-wirelines becomes large enough to affect the impedance quite appreciably,even within the voice frequency range, while enormously affecting it atvery low frequencies.

The general nature of the effect produced on the characteristicimpedance K by any value of the lcakance G is readily seen from mereinspection of equation (1), so far as regards the absolute value andangle of the impedance. Thus, increasing G from any negative and has thevalue %tan",:;; in

creasing G decreases this ne ative angle until G has become as large asC/L, when the angle has become zero and the impedance has become equalto the simple value /L/0, and thereby equal also to /R/G'. Increasing Gbeyond this transition value RC/L toward infinite values gives totheimpedance a positive angle which continually increases wL F whlle theabsolute value of the impedance goes on continually decreasing towardits limiting value of zero. Y

The statements in the-foregoing paragraph hold at all frequencies,though. the effects of leakance are usually most pronounced at lowfrequencies. In factat zero frequency the characteristic impedance of aline having any finite leakance however small is merely lR/G; and atfrequencies so low wL is small compared with R and 100 small comparedwith G, the impedance K is, approximately,

fltl tat i and hence is at least roughly equal to JR/G.

Of course with actual lines this whole physically possible range ofvariation of G from zero to infinite values is never traversed. On thecontrary the leakance G even in open-wire lines seldom reaches a valueas toward its limiting value tan" large as the transition value RC/L andhence the angle seldom becomes positive; while in cables the angleprobably always remains negative and indeed is at least roughly-equal toits limiting value of tan except at very low frequencies.

Impedance of open-we're lines.

It will be recalled that the characteristic impedance of an ordinaryopen-wire depends primarily on its inductance and its capacity, onlysecondarily on its resistance, and far less still on its leakance; andhence that its impedance is at least roughly equal-to /L/O.

Of the quantities defined in equations (2) open-Wire lines are F, k, b,and a. F is suitable as the independent variable, approximatelyproportional to the frequency. is is suitable as one parameter. For theother parameter, which evidently must involve the leakance, b or arespectively is the most suitable according as the leakance G isapproximately proportional to the frequency or is approximatelyindependent of the frequency.

nvraeos The corresponding suitable forms of the equation for thecharacteristic impedance K are then The quantity k dm which occurs in(8) and. (9) as a mere factor is significant as being the value that theimpedance approacheswhen the frequenc is indefinitely increased,provided that G/f approaches zere; it is also the value the impedancewould have at all frequencies if, without changing L and C, the linecould be rendered non-dis sipative. For ordinary open-wire lines atvoice frequencies (R/wL small or fairly small compared to unity) it isat least a rough approximation to the value of the impedance.

This limiting value 7c= will be termed the nominal impedance or, morefully the nominal characteristic impedance.

(Strictly speaking, is varies slightly with the frequency, because ofthe variations of L and even C.)

- The amount K k by which the characteristic impedance K exceeds thenominal characteristic impedance is will be termed the excess impedanceand hence 1ts two components the excess reslstance and the excessreactance; (or, more fully, for the three the excess characteristicImpedance, excess characteristic resistance, and excess characteristicreactance, respectively). The latter two have respectively the values Mk and N, since is is real; M and N denoting the resistance and reactancecomponents of K. The concept excess impedance will be found convenientin various connections, particularly.

imulating form. corresponding to (8) is to (7) the four most suitablefor describing 1+iF W of whichthe components are' 1 +b)F 1bF zu+beFaHere and throughout this specification, I

shall use the notation ag z for mmg where 2=m+iy. Accordingly in thepresent instance,

In the form corresponding to (9) we have 1 iF z a 6F (11) /a+ F (1 F(03+ F (1 .a) F 2(u F )o: ag z= %tan i 2,f

Thus a, which is proportional to the characteristic impedance K (exceptfor the fact that the proportionality factor k is not strictlyindependent of the frequency), depends merely on the two quantities Fand b, or F and a, and hence can be readily represented by tables orgraphs.

When 2 has once been tabulated or graphed the value of K in any specificcase (R, G, L, C specified) is readily obtained therefrom by enteringsuch table or graph of a with the values of the arguments F=mL/R andb=G/wC of (10) or the arguments F=wL/R and (1 GL/RC of (11) and thenmultiplying the value of 2; there found by 7c /L/C. (Graphically thiswould amount merely to a change of scales if the parameters employedwere strictly independent of the frequency.) Thus the function z=/(1+?IF)/(b+i)F (1 z'F) ,(a +111 represents simply and comprehensivelythe properties of the character istic impedance of all smooth lines,though it is more suitable for representing open-wire lines than cables.

The two components 01 and y of z are represented as functions of F bythe curves in Figs. 1, 2 and 2 with b and a respectively as parameters.

When there is no leakance (G O, and hence b==0 and 0 0) equations (10)and (11) reduce to the same form, namely The components and angle aregiven by the following equations:

The relation y= /3 1 can be written in which shows that m 1 is alwayssmaller than y; and is very much smaller except at small values of F,where the two approach equality as F approaches zero.

The foregoing limiting form given by equation (12) for the relativeimpedance 2 is rather important because it is comparatively simple andyet is a close approximation for the impedance of most actual linesexcept at very low frequencies (since the effects of normal amounts ofleakance are very small except at very low frequencies). It willtherefore now be discussed with some fullness:

For the case of no leakance the formulae for m and y are graphed in Fig.1, (b=0), and in Figs. 2 and 2 (a=0). If the wires were devoid ofresistance (11 0) a: would be equal to unity and y would be zero.

Thus the effect of wire resistance (in a non-leaky line) is to make atgreater than its limiting value unity by the amount 50-1 (the relativeexcess reslstance), and to introduce a negative value of y (the relativeexcess reactance, which is equal to the relative reactance). Both 9;- 1and 3 increase with decreasing F; the increase being slow at largevalues of F, but more and more rapid as F is decreased. ml is alwayssmaller than 3 and is much smaller except at low values of F, where thetwo approach equality as F approaches zero. The statements regarding wand 1 hold also for the effect of wire resistance on the characteristicresistance and the characteristic reactance, since these are(approximately) proportional to m and 3 respectively, theproportionality factor being the nominal impedanceJL/C Before leavingequation (12) attention will be directed to certain approximate andexact forms of this equation that have been found very useful indevising and proportioning networks for simulating the characteristicimpedance of smooth lines, as will appear more fully in the latter partof this paper. At large values of F, equation (12) yields immediatelythe approximation when a2l and y have approximately the values Comparingwith (13) and (l4) it is seen that each of the approximations (16) and(17) is always somewhat larger than the It will be recalled that theimpedance of an ordinary cable depends chiefly on its capacity andresistance, relatively little on its inductance, and far less still onits leakance; and hence that its impedance is at least roughly equal to/R/iw0= (1 iM/R/QwO.

Of the quantities defined in equations (2) to (7), the four mostsuitable for describing cables are 15,70, 5 and g. E is suitable as theindependent variable, approximately proportional to the frequency. is issuitable as one parameter. For the other parameter, which evidently mustinvolve the leakance, b or g respectively is the most suitable accordingas the leakance G is approximately proportional to or approximatelyindependent of the frequency. The corresponding suitable forms of theequation for the impedance are then (19) and (20) as given below:

arm

the components and angle being given by the equations the components andangle being given by the equations The two formulae (19) and (20) forcables are less simple than the corresponding formulae (8) and (9) foropen-wire lines, because in (19) and (20) neither of the two parametersenters as a mere factor, and hence the number of efiective parameterscannot be reduced to less than two. For mere specific computations this1s not much of a complication; but in graphical representation it isenough to prevent the desired simplicity and compactness, if therepresentation is required to be exact and comprehenslve.

The general nature of the efiect of leakance in smooth lines has alreadybeen indicated under the heading The general effect of leakance. Incables, particularly, the effect of leakance is usually extremely smallexcept at very low frequencies. Hence in the graphical representation offormulae (19) and (20) it will suffice very well to confine ourselves tothe limiting case of no leakance (G=O, and hence b=0 and g=0), whenthese two equations reduce to the same form, namely 1r= W-i E 21 forwhich the components and angles are expressed as follows:

N= -1/2 EM= {M -k which shows that M7e is always smaller than N, thoughthe two approach equality when E approaches 'zero.

The curves in Fig. 3 represent the resistance and reactance components Mand N of K as functions of E with is as parameter.

An alternative mode of representing the characteristic impedance ofcables is suggested by the fact, already mentioned, that the impedanceof a cable is at least roughly equal to Vii/n00, whence its absolutevalue is at least roughly equal to /R/wO'. This sug gests that we studya relative impedance consisting of the ratio of K to /R/w 0, where 4.0denotes any fixed value of w; and that we adopt the ratio o/w as theindependent variable.- In this mode of treatment it will be urposes ofconvenient to employ the quantities w, 7, F

Thus, w denotes the relative impedance to be studied; its real andimaginary components will be denoted by trend 0, so that w=u+iu 1'-denotes the relative frequency-- relative to any fixed frequency f,. Fis one parameter. The other parameter is respectively bnr b' accordingas the leakance G is approximately proportional to or approximatelyindependent of the frequency. It will be noted that b is the same asalready defined by (7) b, is related to b; and F is related to F, whichhas already been defined by (2). The correspondin forms of the equationfor the relative impe ance 'w are These are seen to be of the samefunctional forms as (19) and (20) respectively; with 'w corresponding toK, r to E, F, to and'b to 9 In other respects, however, there are markeddifferences: K is an impedance, while '20 is a pure number, being theratio of K to /R/w ll; E, though approximately proportional to thefrequency, is not a pure number (for it is not dimensionless), while ris a pure number, being the ratio of the general frequency to any fixedfrequency; of the parameters, k is very different from F and b is verydifferent from 9 The fact that (27) and (28) are of the same functionalforms as (19) and (20) respectively renders formally applicable theequations for the components and angles as given directly afterequations (19) and (20).

As already remarked, the effect of leakance in cables is usuallyextremely small except at very low frequencies. Hence in the graphicalrepresentation of formulae (27) and (28) it will suffice for mostpurposes to confine ourselves to the limiting case of no leakance (G=0,and hence b=0 and b,==0) when these two equations reduce to the sameform,

namely 'w= /Fy- 'i/r. (29) This has the same functional form as (21),with w corresponding to K, 1* to E, and F to 10 a circumstance renderingformally applicable the equations for the components and angle givendirectly following equation (21).

The curves in Fi 4 represent the two com- 7 ponents u and v o w asfunctions of r with F as parameter.

Networks for simulating the impedance of smooth Zines.

Under this heading will be described various networks for simulating thecharacteristic impedance of lines and more particularly of smooth lines.Before proceeding to the systematic description of these networks, someof their practical uses will be mentioned. Foremost of these is theiremployment for balancing purposes in connection with 22.- typerepeaters, already spoken of in the introduction. Another application isfor properly terminating an actual telephone line in the field or anartificial line in the laboratory,

usually for electrical testing purposes or electrical measurements onthe lines. In making certain tests on apparatus normally associated witha telephone line, such line may be conveniently represented forimpedance purposes by the appropriate simulating network.

Some of the networks to be shown are potentiallyequivalent inimpedance;but may difier somewhat in cost, space occupied, etc. For thepurpose of this specification any two networks will be calledpotentially equivalent if, when the elements of either network areassigned any arbitrary values, the other network can be so proportionedas to have at all frequencies identically the same impedance as thefirst network. Evidently the mathematical condition for such equivalenceis that the expressions for the impedances of the two networks have thesame functional forms when the frequency is regarded as the independentvariable. The two networks will then have the same number of independentparameters, ordegrees of freedom for adj ustment; and this number is thesame as the minimum number of elements requisite for the construction ofa network to have identically the impedance of the given network.

For most of the networks described, there are included designformulaefor the values of the network elements (resistances and capacities) Butin any applications requiring the highest simulative precisionattainable with such networks, these formula should be rearded merely asfirst approximations servmg to reduce the requisite detailed designworkdown to a relatively small amount but not permitting it t0 be dispensedwith entirely; for the best values of the network elements dependsomewhat on the particular frequency-range involved, and on thereassigned weighting'of the desired simu ative precision with respect tothe frequency. Moreover, these formulae completely ignore leakance;while actually leakance may not always be quite negligible, even in thevoice frequency-range. A supplementary semigraphical method as an aid tofinally arriving est at relatively low frequencies.

at the best proportioning of the networks-somewhat on the specificfrequency-rangeinwill be disclosed at the proper stage in thisspecification.

The basic resistance and the (access-simulator.

The first approximation to a network for simulating the characteristicimpedance K of a smooth line is evidently a mere resistance E, (Fig. 5)approximately equal to the nominal impedance Z: of the line, that is,

1= w and this is a very close approximation, for instance, in the caseof open-wire lines at the frequencies of carrier current transmission.Compare equation (4) and see the foregoing paragraph following equation(9). The derivation of formula (30) is given below in connection withthe derivation of formula (31). The resistance having the value R, ofthe foregoing equation (30) is the basic resistance.

Over the voice frequency range, however, a mere resistance does notsuffice; since there the excess characteristic impedance K-k is notnegligible, particularly at the lower frequencies. But the resistance R,equal to the nominal impedance may be retained as the natural basis of anetwork if it is supplemented by an element or elements such as toapproximately simulate the excess characteristic impedance. Such asupplementary network is here termed an excess-simulator, and issymbolized abstractly by Fig. 5; while Fig. 5: represents thecorresponding complete net- I work consisting of the basic resistance R,in series with the excess-simulator, whose impedance is denoted by J. Inpractice the term low frequency corrector instead of excess-simulatorhas become rather firmly established. It was suggested by the fact thatthe excess impedance to be simulated is larg- The requisiteexcess-simulator is obviously less simple in structure and proportioningthan the mere basic resistance; whence most of the remainder of thisspecification will be concerned with various specific types ofexcess-simulators.

The simplest excess-simulator, and complete network.

The simplest type of excess-simulator is a mere capacity C, (Fig. 6 Thisis adequate only for those lines whose excess characteristic resistanceis negligible; as, for instance, large gauge open-wire lines, and eventhen not at Very low frequencies. The capacity C, is capable ofsimulating the reactance N of such a line rather closely, and its propervalue for that purpose is approximately although the most suitable valuedepends volved. The complete network (Fig. 6) thus consists merely of aresistance R, and a capacity C in series with each other, havingapproximately the values expressed by (30) and (31);

The network in Fig. 6* consisting of a resistance R, and capacity C, inseries'with each other and having the values expressed by equations (30)and (31) was originally arrived at by working with values of F large orat least fairly large compared with unity; for then, by equation (12)the characteristic impedance K has approximately thevalue Thisrepresents K as having a resistance component k that is independent offrequency, and a reactance component k/2F that is negative and inverselyproportional to the frequency f (since F =wL/R) and thusleads exactly tothe values of R and C expressed by (30) and (31), whence the impedanceof this network is exactly equal to the approximate value of the lineimpedance expressed To obtain more precise and comprehensive knowledgeregarding the simulative preci-.

sion of this network its exact impedance 7cilc/2F will here be comparedwith the exact value of the line impedance (when leak age is neglected).For this purpose it is convenient to employ the line impedance in theform K=wlcik/2mF, (31),

i /m, 1) ax a E, (31

which differ only by the factor a: from the values of R and C expressedby (30.) and (31). Thus the ideal resistance R, and capacity C, forexactly simulating the line impedance would vary with F in precisely thesame way as w varies with F. Moreover the ratio of these ideal values tothe fixed values of R and C, expressed by (30) and (31) is merely 00. Byreference to Fig. l (with 6 0) it will be seen that, except at smallvalues of F, the factor m is merely independent of F and is onlyslightly greater than unity. Thus the values of R and C, determined bymeans of equations (30) and (31) Since (31) can, by (30), be written inthe form l and since an is always greater than unity, it is seen thatthe simulation can be somewhat improved. by supplementing theexcess-simulator with a small series resistance element R the idealvalue of which would be Actually, since a: varies with frequency, R islimited to some compromise value. In practice R would usually becombined with J the basic resistance'R though the functions of the twoare distinctly different. (If the requisite value of R were negative, Rwould merely be decreased by that amount.)

The majority of present-day applications require such high slmulati-vepreclsionthat the excess characteristic resistance of the line is notnegligible, and also a mere capacity does not in all cases simulatetheexcess character-istic reactancequite as closely as desirable. To meetthese needs there have been devised the more precise, yet fairly simple,excess-simulator and complete networks described under several of thefollowing headings.

Two precise types of ewcess-simulators,'and

' their limiting for me.

Figs. 7 and 7* represent two potentially equivalent excess-simulatorsthat in most cases admit of such proportioning as to simulate with therequisite high precision the excess characteristic impedance of theline; the complete network then consisting of either of theseexcess-simulators in series with the basic resistance element R, of Fig.5.

In an specific application the most suit-' able vaf ues for the elementsconstituting either of the two excess-simulators in Figs. 7 and '2'depend somewhat on the particular frequency-range involved, and also onthe weighting of the desired simulative-precision with respect to thefrequency. As might be expected, therefore, the work of determiningclosely the best combination of values for the elements of theexcess-simulator can hardly avoid a certain amount of tentative detaileddesign-work; but usually this can be reduced to a relatively smallamount by some such semi-graphical method as outlined hereinafter.Moreover, first-approximation values that will usually prove to berather close, canbe quickly found by means of equations (32) to (37)which are approximate design-formulae and which are explicit except forcontaining the single undetermined parameter D, whose significance willbe pointed out presently. These formulae are such that theexcess-simulator will possess high simulative-precision at la rge andeven fairly large values of F, for all physically admissible values of D(-0 5 D5 1) and at the lower values of F will have a considerable rangeof adjustment by means of D, whose optimum value can be readilydetermined from inspection of Figs. 8 and 9, as described below. Theabove-mentioned approximate design formulae for the elements of the twoexcess-simulators in Fig. 7 are:

When the excess-simulator is proportioned 1D. accordance with thesedesign formulae the corresponding complete network consisting of suchexcess-simulator J in series with the basic resistance R /lT/@ willpossess the simulative precision represented by the set of graphs inFig. 8, which shows the percentage impedance departure 8 of the completenetwork R,+J from the line impedance K, as function of F with D asparameter. In any specific case, where, of course, the F-range would beknown, inspection of these graphs (Fig. 8) enables the best value of Dto be readily determined, and the correspondin resulting precision 8 tobe seen as function 0 F. The curves show that the best value of D isdetermined by the lowest value of F contemplated, since the departure 8is largest at small values of F .and rapidly decreases toward the largervalues of F. It will be noted that the curves for the limiting valuesD=0 and D l have been included in Fig. 8; the corresponding limitingforms of the excess-simulators are considered a little further on.

The two sets of formulae (32), (33), (34:) and (35), (36), (37)representing first approximations to the proper values of the elementsconstituting the excess-simulators in Figs. 7 and 7 respectively, wereoriginally obtained by working with values of F large or at least fairlylarge compared with unity;

for then, by (15), the excesscharacteristic impedance Klc hasapproximately the value k. .k L on the impedance J =P +z'Q, of eachexcess-simulator in Figs. 7 a and 7 b can be expressed approximately bythe equation derived from the exact equation (37) below, in which t, Pand T have the values ,readily from the two sets of defining e definedby the following two sets of equations (37), (37), (37) and (37), (37),

(37'?) for the excess-slmulators in Figs. 7 and 7 b respectively:

P thus being the value of at w=0. Comparison of the approximateequations (37 and (37") gives immediately as the two conditions that arenecessary and sufficient for (approximate) equality of J and Klc atlarge values of F and 'I. This.

pair of equations is equivalent to the more convenient equatlons,

Li f5 F I 1 t 7:

Thus the ratio of T to F is fixed as soon as either t or P /k is fixed.It will be convenient to adopt {P /2k as the arbitrary quantity and todenote it by D, so that D /11/27: (37 whence P =2D%, (37 and and T =4DF.(37) tions (37), (37), (37) and (37), (3 (37") respectively. The formulafor plotting the curves in Fig.

which is thus the exact formula for the relative impedance J flc of eachof the excess-simulators in Figs. 7 and 7 b when these are proportionedin accordance with the formulae (32) to (37). l 7

A semi-graphical method for proportioning the excess-simulators in Figs.7 and 7 will now be outlined. In this method the ratio T/f is offrequent occurrence and will be denoted by d. Then, recalling thatP+iQ=J, it will be seen from equation (37) that P/P depends only on fand (1; while Q/P depends on f, d, and 27. These observations are thebasis for the method now to be described for evaluating the threeparameters P d, and t which implicitly determine the elements of theexcess-simulators in Figs. 7 and 7 In the first step of this method thetwo parameters d and P are so chosen that the resistance component P ofthe excess-simulator will be approximately equal to the excessresistance Mk of the line impedance K, over the specific f-rangecontemplated, or else will differ therefrom by a nearly constant amount,which can be approximately simulated by a mere series resistanceelement.In the second step of the method the remaining parameter, t, is sochosen that the reactance component Q of the excess-simulator will beapproximately equal to the reactance' N of the line impedance, when dand P have the pair of values already chosen in the first step. Thetechnical procedure in these two steps may now be formulated moreprecisely as follows:

First, over the contemplated f-range, plot a set of curves representingP/P as function of. f with d as parameter; and on the same sheet a setof curvesrepresenting (Mk) /P as function of f with P as parameter. Toevaluate d and P choose (by interpolation,

if necessary) such P/P -curve and (M'k) P curve as most closelycoincide. A preliminary idea regarding the useful ranges of d and P canbe readily obtained from the approximate formulae (37) and (37),t0getherwith Fi 8. v gecond, on another sheet plot as function of j thatparticular N/P -curve having as parameter the value of P already foundin the first step. With this value of P and the correspondmg value of d,as found in the first step, plot also a suificient set. of Q/P -curvesas function of f with tas parameter to find 'the one that coincides mostclosely with the single N/P -curve already plotted. To abridge this steptentative values for t can be readily obtained from the approximateformula (37), together with Fig. 8. But the useful range of t can bedemarcated more closely by solving for t the equation obtained byequating the expressions for Q, and N; the value for 25 thus found is Itwill be recalled that the line-reactance N is practically alwaysnegative. The values of It may even be plotted, as function of f, to seewhether the requisite value of it varies much in the contemplatedf-range.

If the best compromise value of t found 1n the second step isunsatisfactory as regards simulation of N by Q,it will be necessary torevert to the first step, choose some other pair of values for d and Pand with these repeat the second step. In this connection 1t should benoted that, in the first step, it is not necessary to choose the P /P-curveand (M-k) /P -curve which most closely coincide; on the contraryit suflices to choose two curves that are closely parallel (that is,have closely equal slopes at each 7). For, corresponding to the nearlyconstant distance between such two curves, it will only be neces sary tosupplement the excess-simulator with aseries resistance element E -whichwill thus in the complete network be also in series relation to thebasic resistance R and hence can be merged therewith (even when therequisite R is negative, provided it is less than R, in absolute value).v

After the parameters t, P and d=T/f have been evaluated, the values forthe elements of the excess-simulators in Figs. 7 and 7 can be readilyobtained from the two sets of equations (37), (37), (37) and (37), (37 g(37 respectively; it thus being found that The requisite value for thesupplementary series resistance element R is evidently which will beapproximately independent of f if the curves ofP/P and (Mlc)P chosen inthe first step are approximately parallel.

If the requisite value of R is negative, the

F, the resulting network will have at that F exactly the departure shownon Fig. 9, but at all other values of F will, of course, have departureslarger than those on Fig. 9.

It should be noted that these statements regarding the departurespertain to the network when the excess-simulator is proportioned inaccordance with formulae (32) to (37). As those are only firstapproximation formulae, the ultimate precision attainable will usuallybe better, and may be adjusted to possess a somewhat differentdistribution over the frequency-range.

Although the two excess-simulators in Figs. 7 and 7 are potentiallyequivalent as regards impedance there is a slight choice between themfrom the viewpoints of cost and space occupied. For it is readily seen bmore inspec- Figs. 10 and 10 represent the two limiting forms of theexcess-simulators in Fig. 7, corresponding to the limiting values 0 and1 of the parameter D occurring in the design equations (32) to (37). ForD=0 the limiting form is that in Fig. 10, and will be recognized as thesimple excess-simulator already shown in Fig. 6 consisting of a merecapacity (L, having the value expressed by (31) while for D=1 thelimiting form is that in Fig. 10 The departure-curves for these twolimiting forms (D=0 and D=1) are included in Fig. 8, as alreadymentioned; and from them it is seen that the form in Fig. 10 (D=1)possesses much i her simulative precision than the form in ig. 10(D=O)--as would be expected.

For the limiting form of excess-simulator in Fig. 1O thedesign-procedure is considerably simpler, because the parameter t isfixed (t=0). The two remaining parameters (1 and P can be evaluated byinspection of two sheets of curves plotted as functions of 7: One sheetcontainlng a set of curves of P/R, with d as parameter, and curves of(M7c) P with P as parameter; and the other sheet, curves of Q/P with das parameter, and

curves of N /P,, withP as parameter. F our precise types of completenetworks, and

their limiting forms. Figs. 11 and ll 'represent the two poten tiallyequivalent complete networks that can be constructed from the basicresistance R, of Fig. 5*, and the excess-simulators in Figs. 7 and 7 brespectively; and hence having for their elements approximately thevalues expressed by equations (30-), to (37).

Figs. 11 and 11 represent two other complete networks that also arepotentially equivalent to those in Figs. 11 and 11".

The following three sets of formulae express the Values of the elementsconstituting -the networks in Figs. 11, 11, 11 respectively in terms ofthe elements constituting the network in Fig. 11 when those fournetworks have equal impedances. These formulae involve the two ratiosand pertaining to the network in Fig. 11 and defined by the equationsaim +5)" 1 r)" enamel,

Figs. 11 11, 11, 11, are potentially equiva lent as regards impedancethere is some choice among them from the viewpoint of cost and spaceoccupied. For it is readily seen by mere inspection of the networks atzero frequency that, when they have equal impedworks at infinitefrequency it is seen that the Gs being the reciprocals of theRs and thusbeing'the corresponding conductances. Before leaving Figs. 11, 11, 11and 11 it may be noted that the network in Fig. ll has the same form asthough obtained by connecting in parallel two networks having the sameform as Fig. 6 but with elements R C, and R C say. Now it is known that,in most applications, the network in Fig. 11 has much higher simulativeprecision than that in F ig..6. These considerations suggest thepossibility of attaining still higher precision by connecting inparallel several such networks, having constants R1I.C1/; Bill 61!;R111, G1!!! I F1gs.-12, 12 and 12 represent the two limiting forms ofthe networks in Figs. 11, 11*, 11 and 11, corresponding to the limitingvalues 0 and 1 of the parameter D occurring in the design-equations (32)to (37). For D=O the limiting form is that represented in Fig. 12, andthis will be recognized ,as the simple 2-element network already shownin Fig. 6; while for D=1 the limiting forms are the two potentiallyequivalent 3-element networks represented in Fi s. 12 and 12.

The values of the elements 0 the network in Fig. 12 in terms of theelements of the net- 'work in Fig. 12*, for equivalence of these twonetworks as regards impedance, are

R =R +R =3R (41) R =R ('1+R /R )=3R /2, a 1 2 Thus the network in Fig.12 requires only four-ninths as much capacity as the network in F ig. 12

Modifications for very Zow frequencies. Thus far the presentspecification has dealt with the characteristic impedance of smoothlines as distinguished from their sending end impedance, strictlyspeaking. The two are closely equal when the lines are electricallylong, which is usually the case for the tele o5 Although the fourcomplete networks in phonic frequency range; but at very low frequenciesthe sending end impedance of even a rather long line may depend verygreatly on the distant terminating impedance and hence depart widelyfrom the characteristic lmpedance. In case the terminating impedance isconductive to direct current the sendmg end impedance of even a strictlynonleaky line would have a finite value at zero frequency; itsresistance component evidently being equal to the total line wireresistance plus the terminating resistance, while its reactancecomponent would, of course, be zero. Actually, on account of lineleakance, the resistance component would be somewhat less; and in casethe distant terminating impedance permits no passage of direct currentthe sending end impedance of the line at zero frequency would dependlargely on the line leakancer Most of the simulating networks thus fardescribed were devised primarilywith regard to the voice range offrequencies, without reference to frequencies very far below that range.At very low frequencies these networks become unsuitable because theirimpedance is not only much too large but also has not even approximatelythe proper angle. There have not been many occasions for modifying thenetworks so as to extend their range of simulation down toward zerofrequency; but it seems likely that in most cases the requisitemodification in the network impedance could be attained,'at leastroughly, by shunting the excess-simulator (Fig. 5")

with a mere resistance S approximately equal to the zero-frequencysending end resistance of the line diminished by the resistance R of thebasic resistance element. Clearly this modification will give the net-'work the desired impedance at zero-frequency, without affecting itsimpedance, at

infinite frequencies; since the impedance of the unshuntedexcess-simulator 1s infinite (except for the limiting form in Fig. 10)",at

zero-frequency and is zero at infinite frequencies. At the intermediatefrequencies the resulting modification would doubtless be slight excepttoward the lower frequencies, where it would increase more and morerapidly as zero-frequency is approached. Of course the addition of themodifying element S would usually entail some alterations in theproportioning of the original network, as indicated by Fig. 13, wherethe altered values of J and R are denoted by J and R respectively. a

Fig. 13" represents an alternative but potentially equivalent form ofmodification, obtained by shunting the original form of network (Fig. 5)with a resistance S"; and the conditions for equivalence are 7 beenstated only with reference to the excess-simulator regarded abstractly.When the specific structure of the excess-simulator is regarded, themodification can take several different forms which, for any oneexcess-. simulator, are equivalent as regards impedance. Certain ofthese are noted in the following paragraphs:

Among the modified excess-simulators will evidently be found one havingthe limiting form already depicted in Fig. 10".

Figs. 14 and 14" represent respectively,

the S-element excess-simulators in Figs. 7

and 7 modified by the shunt resistances, and thereby converted to4-element excesssimulators. Figs. 14 and 14 represent two otherit-element excess-simulators that are potentially equivalent to those inFigs. 1? and 14 as regards impedance.

It may be noted that the excess-simulator in Fig. 14 has the same formas though obtained by connecting in series two of the simple form ofmodified excess-simulator in Fig. 10 having elements 0 R and C B say.This observation suggests the possibility of attaining still highersimulative precision for a modified excess-simulator by connecting 1nseries several such slmple modlfied excess-simulators.

Among the modified complete networks will evidently be found two havingrespectively the forms already depicted in Figs. 12" and 12.

Figs. 15, 15", 15 and 15 represent four potentially equivalent completenetworks derived from Fig. 11 by application of a shunt resistance S".The forms in Figs. 15 and 15", though each containing a su erfluouselement, are of interest because t ey have the same forms as thoughobtained by connecting in parallel two networks of the forms alreadydepicted in Figs. 12 and 12" respectively. Fig. 17 shows the entireunits of Fig. 11 in parallel.

Modifications for leaky lines.

For lines whose leakance is not quite negligible a study of the formulaeand graphs of the line impedance indicates that the effects of suchleakance can be sufficiently taken into account by a mere slightreproportioning of the network without the addition of any furtherelement, except that a small series inductance might be a slightimprovement in those cases where the leakance increases rapidly with thefrequency; this modification is shown in Fig. 16.

I lbustmtive ewample. The example contained in this section serves twopurposes. First, it illustrates the use of two types of the general lineimpedance graphs contained in the sections entitled Impedance of openwire lines and Impedance of cables; second, it illustrates the firstapproximation dcsign'of a simulating network by means of the'method inthe part of the specification following'those sec tions.

The specific example chosen pertains to a well-insulated open-wire lineconsistmg of two horizontal parallel copper wires, of No. 12 N. B. S.gauge, liaving per loop mile the constants R=10A ohms L=.00367 henrles.G=.00835 X 10' farads, the leakance G being regarded as negligible. Thisparticular type and gauge-offline was chosen because it is ratherextensively employed in practice, and

also because its excess impedance is far from being negligible even asregards its resistance component.

For the illustrative purposes contemplated, it will be supposed that itis desired to evaluate the resistance and reactance components M and Nof the characteristic impedance K of this line over the frequency-rangefrom 200 to 2500 cycles per second; and also to design a network forapproximately simulating this impedance over that frequencyrange, and todetermine the simulative precision of such network.

The procedure and results are indicated by the following table togetherwith the supplementary description coming thereafter.

I F x -u M --N r u a an 6 200 .444 1.32 .86 870570 .222 1.87 1.22 3.0300 .666 1.19 .64 790 425 .333.1.60 .91 16.6 1.3 500 1.110 1.08 .42 717270 .555 1.53 .60 8.1 .3 800 1.776 1.04 .27 691 179 .888 1.48 .38 3.6 .11200 2.064 1.02 .10 676 126 1.332 1.45 .27 1.7 .1 1600 3.552 1.01 .14670 93 1.770 1.43 .20 1.0 .1 2000 4.440101 .12 670 80 2.220 1.43 .17- .6.1 2300 5.100 1.01 .10 670 66 2.553 1.43 .14 .5 .1 2500 5.550 1.01 .10670 66 2775 1.43 .14 .4 .1 w m 1 0 668 o m /2 0 0 0 The first column inthe table is a set of values of the frequency f distributed over thespecified range 200 to 2500.

The columns headed F, a -g M, N show the successive steps in evaluatinthe characteristic impedance K=M+7IN y means of Fig. 2, with a=0 (sincethe leakance G is neglected). Or Fig. 1 may be used, with b=0;' .butFig. 2 is plotted to a larger scale. The

F-column was obtainedfrom the f-column by means of the equation F=eL/R=.00222f. Next the values of w and 3 were read from the curves a=0 inFig. 2. Finally M and N were gained from.M=ka: and N=kg,/, with Zc=/L/0= 663 ohms whence M and N are in ohms). From the table it will benoted that at f=200 the excess resistance is about one third as large asthe nominal impedance. and the reactance is about nine-tenths as largeas the nominal impedance.

The colums r, 'u, 12 show the steps in evaluating M and N by means ofFig. 4. F (in Fig. 1) is chosen equal to 2. F =1 would be somewhatpreferable for reading ofli values; but the principles are more clearlyexhibited by choosing for F some other value than unity. The r-columnwas obtained from the f-column by means of the equation 1== f/f=27rLf/RF =.O0111f. Next the Values of uand 1; were read off from thecurves F =2 in Fig. 4. Finally the values of M and N were obtained fromll1= [K 171 andN IKIIU,

The last two columns (8 0) of thefable show the percentage precision ofcertain simu1-' fication, and having for their elements the values givena little further along in this' specification. v

8 pertains to the simple 2-element network in Fig. 6 when proportionedin accordance with equations (30) and (3 The values of 8 were read fromthe curve D=0 in Fig. 8. The precision at the lower frequencies could beconsiderably improved by the addition of a small resistance R as alreadynoted in connection with equation (31), but with a correspondingsacrifice in the rest of the range.

8 pertains to the 4-ele1nent networks in Figs. 11 and 11 whenproportioned in accordance with (30), (32), (33), (34) and (30), (35),(36), (37 respectively; andSpertains also to the networks in Figs. 11and 11 when these are proportioned by the formulae heretofore given forthe elements of those figures, so as to be equivalent to the-network inFig. 11. The values of 8 were read from the curve D=O.55 in Fig. 8,thisvalu e of D being known from Fig. 9 to be about the best.

The valuesof the elements of the networks to which 8., and 8, pertainwill now be set down, each responding diagram and design-.formulae:

Fig. 6; formulae (30), (31) precision 8 Fig. 11 formulae (30), (32),(33), (34); precision 8.

R =663 ohms, O =1.063 10- farads,

farads,

charts would evidently have to be drawn atmuch closer intervals than hasbeen done in the drawings with this specification, Where the purpose ofthe charts is mainly qualitative, or only roughly quantitative, toexhibit the general nature claim:

1. The method of increasing the precision of the functions involved.

preceded by a reference tothe corof simulationat low frequencies ofcurrent between a network of lumped elements and a transmission line,which consists in shunting part of the current around saidnetwork withtheinverse ratio of this shunted current to its electromotive force at areal number value substantially higher than the value of the basicresistance of said line.

2. A network of lumped impedance elements to simulate the impedance of atransmission line over a wide frequency range, in combination with ashunt resistance of high value compared to the basic resistance for saidline to improve the simulation at low frequencles.

3. The method of making a network have I the same impedance .as atransmission line at low frequencies and .at other frequencies whlchconsists in sending the current pr1nc1- pally through a shunt resistanceat the low frequencies and principally through the main part of thenetwork at said other frequencies.

4. As a' device for simulating the impedance of a transmission line overa frequency range comprising lowfrequencies, a network comprising aseries resistance corresponding approximately to the impedance of theline, a condenser as an excess impedance simulator in series with saidresistance and a further resistance shunting both said condenser andsaid first-mentioned resistance.

5. As an excess impedance simulator, a network comprising a condenserand another conderiser'and a resistance in parallel with simulate saidimpedance, the combined effect of said plurality of networks withindividual adjustment thereof being to give a close simulation of thesaid impedance. 7 In combination, an extended transmission line in whichthe leakance increases rapidly with the frequency, and an artificialline to simulate it, said artificial line con-. sisting of lumpedelements combined to simulate t having added thereto a small seriesinductance.

In testimony whereof, I have signed my name to this specification this5th da of April, 1923. RAY S. HO

e extended line without leakance and

